3.19 \(\int \frac{\left (b x+c x^2\right )^{3/2}}{x^5} \, dx\)

Optimal. Leaf size=23 \[ -\frac{2 \left (b x+c x^2\right )^{5/2}}{5 b x^5} \]

[Out]

(-2*(b*x + c*x^2)^(5/2))/(5*b*x^5)

_______________________________________________________________________________________

Rubi [A]  time = 0.0290212, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ -\frac{2 \left (b x+c x^2\right )^{5/2}}{5 b x^5} \]

Antiderivative was successfully verified.

[In]  Int[(b*x + c*x^2)^(3/2)/x^5,x]

[Out]

(-2*(b*x + c*x^2)^(5/2))/(5*b*x^5)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 3.66271, size = 20, normalized size = 0.87 \[ - \frac{2 \left (b x + c x^{2}\right )^{\frac{5}{2}}}{5 b x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x)**(3/2)/x**5,x)

[Out]

-2*(b*x + c*x**2)**(5/2)/(5*b*x**5)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0329845, size = 21, normalized size = 0.91 \[ -\frac{2 (x (b+c x))^{5/2}}{5 b x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x + c*x^2)^(3/2)/x^5,x]

[Out]

(-2*(x*(b + c*x))^(5/2))/(5*b*x^5)

_______________________________________________________________________________________

Maple [A]  time = 0.005, size = 25, normalized size = 1.1 \[ -{\frac{2\,cx+2\,b}{5\,b{x}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^(3/2)/x^5,x)

[Out]

-2/5/x^4*(c*x+b)/b*(c*x^2+b*x)^(3/2)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.217579, size = 47, normalized size = 2.04 \[ -\frac{2 \,{\left (c^{2} x^{2} + 2 \, b c x + b^{2}\right )} \sqrt{c x^{2} + b x}}{5 \, b x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)/x^5,x, algorithm="fricas")

[Out]

-2/5*(c^2*x^2 + 2*b*c*x + b^2)*sqrt(c*x^2 + b*x)/(b*x^3)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x \left (b + c x\right )\right )^{\frac{3}{2}}}{x^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x)**(3/2)/x**5,x)

[Out]

Integral((x*(b + c*x))**(3/2)/x**5, x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.214912, size = 181, normalized size = 7.87 \[ \frac{2 \,{\left (5 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} c^{2} + 10 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} b c^{\frac{3}{2}} + 10 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} b^{2} c + 5 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} b^{3} \sqrt{c} + b^{4}\right )}}{5 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)/x^5,x, algorithm="giac")

[Out]

2/5*(5*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*c^2 + 10*(sqrt(c)*x - sqrt(c*x^2 + b*x)
)^3*b*c^(3/2) + 10*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*b^2*c + 5*(sqrt(c)*x - sqrt
(c*x^2 + b*x))*b^3*sqrt(c) + b^4)/(sqrt(c)*x - sqrt(c*x^2 + b*x))^5